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A093915 Triangle with r-th row containing r consecutive integers that sum to the smallest possible proper multiple of A006003(r).

%I A093915 #12 Jan 17 2020 14:04:53
%S A093915 2,7,8,9,10,11,24,25,26,27,24,25,26,27,28,53,54,55,56,57,58,47,48,49,
%T A093915 50,51,52,53,94,95,96,97,98,99,100,101,78,79,80,81,82,83,84,85,86,147,
%U A093915 148,149,150,151,152,153,154,155,156,117,118,119,120,121,122,123,124,125,126,127,212,213,214,215,216,217
%N A093915 Triangle with r-th row containing r consecutive integers that sum to the smallest possible proper multiple of A006003(r).
%C A093915 The r-th row constructed as explained in the example starts with x=A093916(r), ends with x+r-1=A093918(r), and has its sum A093917(r) equal to the smallest proper multiple of A006003(r). There is a simple formula for A093917(r), which allows us to calculate A093915(n) directly. - _M. F. Hasler_, Apr 04 2009
%e A093915 Given the triangle
%e A093915 1 . . . . with row sum S1 = 1 = A006003(1)
%e A093915 2,3 . . . with row sum S2 = 2+3 = 5 = A006003(2)
%e A093915 4,5,6 . . with row sum S3 = 4+5+6 = 15 = A006003(3), etc.,
%e A093915 the sequence is constructed as follows:
%e A093915 The first row below must be a proper (i.e., > 1) multiple of S1; the smallest possibility is [ 2 ].
%e A093915 The next row below must contain 2 consecutive integers with sum equal to a proper multiple of S2=5. It cannot be 10 (not the sum of 2 consecutive integers), but 15 = 7+8 is a possibility.
%e A093915 The third row [x,x+1,x+2] must sum to a multiple of S3=15, and 2*S3=30 is possible for x=9.
%e A093915 The 4th row [x,x+1,x+2,x+3] must have its sum 4x+6 equal to a multiple of S4=7+8+9+10=34, and x=24 gives the sum 102=3*34, while 2*34=68 can't be achieved for any integer x.
%e A093915 This gives:
%e A093915 2 . . . . . . . with row sum 2 = 2*S1
%e A093915 7,8 . . . . . . with row sum 7+8 = 15 = 3*S2
%e A093915 9,10,11 . . . . with row sum 9+10+11 = 30 = 2*S3
%e A093915 24,25,26,27 . . with row sum 24+25+26+27 = 102 = 3*S4.
%o A093915 (PARI) for(r=1,9, x=A093916(r)-1; for(c=1,r, print1(x+c,","))) /* _M. F. Hasler_, Apr 04 2009 */
%Y A093915 Cf. A093916, A093917, A093918.
%K A093915 nonn,tabl
%O A093915 1,1
%A A093915 _Amarnath Murthy_, Apr 25 2004
%E A093915 Edited and extended (values beyond a(15), example, PARI code) by _M. F. Hasler_, Apr 04 2009